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I’ve been reading up on Dan Meyers “Ladder of Abstraction.” It turns out he didn’t invent it. I think the phrase was first used by H. I. Hayakawa. And there are obviously different versions of it. The most concrete examples deal with language. (Yes it’s ironic I used the word concrete. One could argue that Hayakawa’s entire ladder is on the lowest rungs of the math ladders.)

Abstracting from Cow to Wealth:

English teachers have it so easy… Except for the reading essays part.

Another version was described by Michael Matassa and Frederick Peck at the ICME-12 conference. It refers to the Iceberg Metaphor and shows students going three stages: Concrete, Preformal, Formal. (I’m not sure if they call it The Ladder of Abstraction.)

I’ve been trying to think of some universal system for understanding and/or naming the different rungs.

You may ask: doesn’t Concrete, Preformal, Formal count? It’s a start. The problem is that, beyond algebra, most math lives in the formal category. For example, you may have a student that totally understands the equations for projectile motion. But when you try to generalize to all parabolas the student becomes completely lost. Both of these live in the “formal” section of the ladder but the second is clearly on a higher rung.

I’ve been trying to think of a way to break it down into more explicit rungs on the ladder. But as pointed out by Dan, it depends on the question you’re asking.

He starts with the same picture and shows how it can lead to different ladders with different rungs. So instead of trying to create a universal ladder it might be more useful to see how the idea of the ladder works with different problems.

So I’m in the process of creating different ladders which all start with a cow as the bottom rung.

While you wait in anticipation you can check out this version which looks at design…